# Matrix Analysis

There is a lot more to matrix analysis than this - see for example SingularValueDecomposition.

and also http://en.wikipedia.org/wiki/Matrix_theory
If the EigenValue's magnitude is greater than 1, it equals (1 + interest rate) [or 1 + growth rate if that's clearer to you].

If the EigenValue's magnitude is less than 1, it equals (1 - decay rate).

If the EigenValue's magnitude is 1, the system neither grows nor decays. It might flip-flop or cycle, though.

If the EigenValue is a complex number, it indicates a cycling system.

How about combinations of EigenValues? It's not all scaling; what makes multiplication by one matrix do a reflection, and another a rotation?

If the EigenValue is -1, multiplying by it does a reflection (which is also a 180 degree rotation).

If the EigenValue is a negative real number, multiplying by it does a reflection (and a scaling).

If the EigenValue is i, multiplying by it is the same as rotating 90 degrees. After 2 rotations, you will do a reflection. After 4 rotations, you will wrap all the way around. This is a cycle.

If the EigenValue is a complex number, multiplying by it does a rotation (and a scaling). After several rotations, you will eventually wrap all the way around. This is a cycle.